Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems

The main characteristics of any wireless communication system are deteriorated as a result 12 of the effect of the additive and multiplicative noise. The effect of addition of noise and 13 interference to the signal generates an appearance of false information in the case of the 14 additive noise. For this reason, the parameters of the received signal, which is an additive 15 mixture of the signal, noise, and interference, differ from the parameters of the transmitted 16 signal. Stochastic distortions of parameters in the transmitted signal, attributable to 17 unforeseen changes in instantaneous values of the signal phase and amplitude as a function 18 of time, can be considered as multiplicative noise. Under stimulus of the multiplicative 19 noise, false information is a consequence of changed parameters of transmitted signals, for 20 example, the parameters of transmitted signals are corrupted by the noise and interference. 21 Thus, the impact of the additive noise and interference may be lowered by an increase in the 22 signal-to-noise ratio (SNR). However, in the case of the multiplicative noise and 23 interference, an increase in SNR does not produce any positive effects. 24


Introduction
The additive and multiplicative noise exists forever in any wireless communication system. 8 Quality and integrity of any wireless communication systems are defined and limited by 9 statistical characteristics of the noise and interference, which are caused by an 10 electromagnetic field of the environment. 11 The main characteristics of any wireless communication system are deteriorated as a result 12 of the effect of the additive and multiplicative noise. The effect of addition of noise and 13 interference to the signal generates an appearance of false information in the case of the 14 additive noise. For this reason, the parameters of the received signal, which is an additive 15 mixture of the signal, noise, and interference, differ from the parameters of the transmitted 16 signal. Stochastic distortions of parameters in the transmitted signal, attributable to 17 unforeseen changes in instantaneous values of the signal phase and amplitude as a function 18 of time, can be considered as multiplicative noise. Under stimulus of the multiplicative 19 noise, false information is a consequence of changed parameters of transmitted signals, for 20 example, the parameters of transmitted signals are corrupted by the noise and interference. 21 Thus, the impact of the additive noise and interference may be lowered by an increase in the 22 signal-to-noise ratio (SNR). However, in the case of the multiplicative noise and 23 interference, an increase in SNR does not produce any positive effects. 24 The main functional characteristics of any wireless communication systems are defined by 25 an application area and are often specific for distinctive types of these systems. In the 26 majority of cases, the main performance of any wireless communication systems are defined 27 by some initial characteristics describing a quality of signal processing in the presence of 28 noise: the precision of signal parameter measurement, the definition of resolution intervals 29 of the signal parameters, and the probability of error. 30 The main idea is to use the generalized approach to signal processing (GASP) in noise in 1 wireless communication systems [1][2][3]. The generalized approach is based on a seemingly 2 abstract idea: the introduction of an additional noise source that does not carry any 3 information about the signal and signal parameters in order to improve the qualitative 4 performance of wireless communication systems. In other words, we compare statistical 5 data defining the statistical parameters of the probability distribution densities (pdfs) of 6 the observed input stochastic samples from two independent frequency time regions -a 7 "yes" signal is possible in the first region and it is known a priori that a "no" signal is 8 obtained in the second region. The proposed GASP allows us to formulate a decision-ma-9 king rule based on the determination of the jointly sufficient statistics of the mean and 10 variance of the likelihood function (or functional). Classical and modern signal processing 11 theories allow us to define only the mean of the likelihood function (or functional). 12 Additional information about the statistical characteristics of the likelihood function (or 13 functional) leads us to better quality signal detection and definition of signal parameters 14 in compared with the optimal signal processing algorithms of classical or modern 15 theories. 16 Thus, for any wireless communication systems, we have to consider two problems -17 analysis and synthesis [8]. The first problem (analysis) -the problem to study a stimulus 18 of the additive and multiplicative noise on the main principles and performance under the 19 use of GASP -is an analysis of impact of the additive and multiplicative noise on the 20 main characteristics of wireless communication systems, the receivers in which are 21 constructed on the basis of GASP. This problem is very important in practice. This 22 analysis allows us to define limitations on the use of wireless communication systems and 23 to quantify the additive and multiplicative noise impact relative to other sources of 24 interference present in these systems. If we are able to conclude that the presence of the 25 additive and multiplicative noise is the main factor or one of the main factors limiting the 26 performance of any wireless communication systems, then the second problem -the 27 definition of structure and main parameters and characteristics of the generalized detector 28 or receiver (GD or GR) under a dual stimulus of the additive and multiplicative noise -29 the problem of synthesis -arises. 30 GASP allows us to extend the well-known boundaries of the potential noise immunity set by 31 classical and modern signal processing theories. Employment of wireless communication 32 systems, the receivers of which are constructed on the basis of GASP, allows us to obtain 33 high detection of signals and high accuracy of signal parameter definition with noise 34 components present compared with that systems, the receivers of which are constructed on 35 the basis of classical and modern signal processing theories. The optimal and asymptotic 36 optimal signal processing algorithms of classical and modern theories, for signals with 37 amplitude-frequency-phase structure characteristics that can be known and unknown a 38 priori, are components of the signal processing algorithms that are designed on the basis of 39 GASP. 40 In the proposed chapter, we would like to present and discuss the following aspects of 1 GASP implemented in the direct-sequence code-division multiple access (DS-CDMA) 2 wireless communication systems: 3  The main theoretical statements and brief description; 4  Signal processing in DS-CDMA wireless communication systems with optimal 5 combining and partial cancellation; 6  Signal processing in DS-CDMA wireless communication systems with frequency-7 selective channels; 8  Signal processing in DS-CDMA downlink wireless communication systems with fading 9 channels; 10  Summary and discussion. 11

CDMA wireless communication systems
(2) 1 where T denotes a transpose. The data distribution in the complex matrix Z can be 2 expressed as: 3 where s E is the average signal energy at the receiver input, and I is the MN MN  identity 5 matrix. We consider a situation when the signaling scheme is unknown (the receiver has a 6 total freedom of choosing the signaling strategy) excepting a known power within the limits 7 of the frequency band of interest. Thus, the receiver should be able to detect a presence of 8 any possible signals satisfying the power and bandwidth constraints for robust detection of 9 the signal ] [k s incoming at the receiver input in wireless communication systems. 10 The generalized receiver (GR) has been constructed based on the generalized approach to 11 signal processing (GASP) in noise and discussed in numerous journal and conference papers 12 and some monographs, namely, in . GR is considered as a linear combination of the 13 correlation detector that is optimal in the Neyman-Pearson criterion sense under detection 14 of signals with known parameters and the energy detector that is optimal in the Neyman-15 Pearson criterion sense under detection of signals with unknown parameters. The main 16 functioning principle of GR is a matching between the model signal generated by the local 17 oscillator in GR and the information signal incoming at the GR input by whole range of 18 parameters. In this case, the noise component of the GR correlation detector caused by 19 interaction between the model signal generated by the local oscillator in GR and the input 20 noise and the random component of the GR energy detector caused by interaction between 21 the energy of incoming information signal and the input noise are cancelled in the statistical 22 sense. This GR feature allows us to obtain the better detection performance in comparison 23 with other classical receivers. 24 The specific feature of GASP is introduction of additional noise source that does not carry 25 any information about the signal with the purpose to improve a qualitative signal detection 26 performance. This additional noise can be considered as the reference noise without any 27 information about the signal to be detected. The jointly sufficient statistics of the mean and 28 variance of the likelihood function is obtained in the case of GASP implementation, while 29 the classical and modern signal processing theories can deliver only a sufficient statistics of 30 the mean or variance of the likelihood function (no the jointly sufficient statistics of the 31 mean and variance of the likelihood function). Thus, the implementation of GASP allows us 32 to obtain more information about the input process or received information signal. Owing to 33 this fact, an implementation of the receivers constructed based on GASP basis allows us to 34 improve the signal detection performance of wireless communication system in comparison 1 with employment of other conventional receivers. The GR flowchart is presented in Fig. 1. As we can see from Fig. 1, the GR consists of three 5 channels: 6  the correlation channel -the preliminary filter PF, multipliers 1 and 2, model signal 7 generator MSG; 8  the autocorrelation channel -the preliminary filter PF, the additional filter AF, 9 multipliers 3 and 4, summator 1; 10  the compensation channel -the summators 2 and 3 and accumulator 1. 11 As follows from Fig. 1 , where s f  is the signal bandwidth, 25 the processes at the AF and PF outputs can be considered as the uncorrelated and 26 independent processes and, in practice, under this condition, the coefficient of correlation 27 between PF and AF output processes is not more than 0.05 that was confirmed 28 experimentally in [22,23]. 29 The processes at the AF and PF outputs present the input stochastic samples from two 30 independent frequency-time regions. If the noise ] [k w at the PF and AF inputs is Gaussian, 31 the noise at the PF and AF outputs is Gaussian, too, owing to the fact that PF and AF are the 32 linear systems and we believe that these linear systems do not change the statistical 33 parameters of the input process. Thus, the AF can be considered as a reference noise 34 generator with a priori knowledge a "no" signal (the reference noise sample). A detailed 35 discussion of the AF and PF can be found in [6,7]. The noise at the PF and AF outputs can be 36 presented in the following form: 37 (4) 1 Under the hypothesis , 1 H the signal at the PF output can be defined as is the vector of random process at the 15 PF output, and GR THR is the GR detection threshold. We can rewrite (5) using the vector 16 form: 17 vector of the random process at the PF output with 19 elements defined as  According to GASP and GR structure shown in Fig. 1 signals are selected according to  1 signal-envelope amplitude [29][30][31][32][33][34][35]. However, some receiver implementations recover 2 directly the in-phase and quadrature components of the received branch signals. 3 Furthermore, the optimal maximum likelihood estimation (MLE) of the phase of a diversity 4 branch signal is implemented by first estimating the in-phase and quadrature branch signal 5 components and obtaining the signal phase as a derived quantity [36,37]. Other channel-6 estimation procedures also operate by first estimating the in-phase and quadrature branch 7 signal components [38][39][40][41]. Thus, rather than N available signals, there are 2N available 8 quadrature branch signal components for combining. In general, the largest 2L of these 2N 9 quadrature branch signal components will not be the same as the 2L quadrature branch 10 signal components of the L branch signals having the largest signal envelopes. 11 In this section, we investigate how much improvement in performance can be achieved improves. There are some papers on the selection of the PCF for a receiver based on the 28 PPIC. In [44][45][46], formulas for the optimal PCF were derived through straightforward 29 analysis based on soft decisions. In contrast, it is very difficult to obtain the optimal PCF for 30 a receiver based on PPIC with hard decisions owing to their nonlinear character. One 31 common approach to solve the nonlinear problem is to select an arbitrary PCF for the first 32 stage and then increase the value for each successive stage, since the MAI estimates may 33 become more reliable in later stages [43,47,48]. This approach is simple, but it might not 34 provide satisfactory performance. 35 In this section, we use the Price's theorem [49,50] We may also assume that the maximum delay is much smaller than the processing gain N 9 [46]. Before our formulation, we first define a L N   ) 1 2 ( composite signature matrix k A in 10 the following form 11 [ ] Since a multipath fading channel is involved, the current received bit signal will be 15 interfered by previous bit signal. As mentioned above, the maximum path delay is much 16 smaller than the processing gain. The interference will not be severe and for simplicity, we 17 may ignore this effect. Let us denote the channel gain for multipath fading as 18

Selection/maximal-ratio combining 20
We assume that there are N diversity branches experiencing slow and flat Rayleigh fading, 21 and all of the fading processes are independent and identically distributed (i.i.d.). During 22 analysis, we consider only the hypothesis 1 H "a yes" signal in the input stochastic process. 23 Then the equivalent received baseband signal for the k-th diversity branch takes the 24 following form: 25 is a 1-D baseband transmitted signal that without loss of generality, is 2 assumed to be real, is the complex channel gain for the k-th branch subjected to 3 Rayleigh fading, k  is the propagation delay along the k-th path of received signal, and 4 is the zero-mean complex AWGN with two-sided power spectral density 2 0 N with 5 the dimension Hz W . At GR front end, for each diversity branch, the received signal is split 6 into its in-phase and quadrature signal components. Then, the conventional HS/MRC 7 scheme is applied over all of these quadrature branches, as shown in Fig.2.

11
We can present ) (t h k given by (29)-(37) as i.i.d. complex Gaussian random variables 12 assuming that each of the L branches experiences the slow and flat Rayleigh fading 13 The GR output with quadrature subbranch HS/MRC and HS/MRC schemes according to 19 GASP [2, 3, 6-9] is given by: 20 is the background noise forming at the GR output for the k-th branch; 22 and 2L of the k b equal 1. 23

Synchronous DS-CDMA wireless communication system 1
Consider a synchronous DS-CDMA system employing the GR with K users, the processing 2 gain N, the number of frame L, the chip duration c T , the bit duration R NT T c b  with 3 information bit encoding rate R. The signature waveform of the user k is given by 4 is a random spreading code with each element taking value on 6 is the unit amplitude rectangular pulse with duration c T . The 7 baseband signal transmitted by the user k is given by 8 is the transmitted signal amplitude of the user k. The following form can 10 present the received baseband signal: 11 where taking into account (29)- (37) and (39) and as it was shown in [13] 13 is the received signal amplitude envelope for the user k, ) (t w is the complex Gaussian noise 15 with zero mean with 16 ); k τ is the 2 delay factor that can be neglected for simplicity of analysis. For this case, we have 3 1, where the first term in (54) is the desired signal; 5 is the coefficient of correlation between signature waveforms of the k-th and j-th users; the 7 third term in (54) 8 is the total noise component at the GR output; and the second term in (54) 10 is the MAI. The conventional GR makes a decision based on k Z . Thus, MAI is treated as 12 another noise source. When the number of users is large, MAI will seriously degrade the 13 system performance. GR with partial interference cancellation, being a multiuser detection 14 scheme [31], is proposed to alleviate this problem. 15 Denoting the soft and hard decisions at the GR output for the user k by 16 respectively, the output of the GR with the first PPIC stage with a partial cancellation factor 18 equal to 1 p can be written by [43] 19 denotes the soft decision of user k at the GR output with the first stage of PPIC 2 and 3 is the estimated MAI using a hard decision. 5

Symbol error rate expression 7
Let k q denote the instantaneous SNR per symbol of the k-th quadrature branch 8 is the mathematical expectation of MGF with respect 9 to SNR per symbol. 10 A finite-limit integral for the Gaussian Q-function, which is convenient for numerical 11 integrations, is given by [54] 12 The error function can be related to the Gaussian Q-function by 14 The complementary error function is defined as which is convenient for computing values using MATLAB since erfc is a subprogram in 1 MATLAB but the Gaussian Q-function is not (unless you have a Communications Toolbox). 2 Note that the Gaussian Q-function is the tabulated function. 3 Now, let us compare (64) and (68) to obtain the closed form expression for the SER of M-ary 4 PAM wireless communication system employing the GR with quadrature subbranch 5 HS/MRC and HS/MRC schemes. We can easily see that taking into account (44), (45), (61), 6 (62), and (65), the SER of M-ary PAM system employing the GR with quadrature subbranch 7 HS/MRC and HS/MRC schemes can be defined in the following form 8 Thus, we obtain the closed form expression for the SER of M- ary are, respectively, the probability density function (pdf) and the 20 cumulative distribution function (cdf) of q, the SNR per symbol, for each quadrature branch, 21 and 22 with pdf given by [49] 3 is the average SNR per symbol for each diversity branch. The MMGF of SNR per symbol of 7 a single quadrature branch can be determined in the following form: 8 Moreover, the cdf of q becomes 10 is the complimentary error function. 12

AWGN channel 14
In this section, we determine the PCF at the GR output with the first stage of PPIC. From 15 [43], the linear minimum mean-square error (MMSE) solution of PCF for the first stage of 16 PPIC is given by 17 is the power of residual MAI plus the total noise component forming at the GR output at the 2 first stage; 3 is the power of true MAI plus the total noise component forming at the GR output (also 5 called the 0-th stage); 6 is a correlation between these two MAI terms. It can be rewritten as 8 where i e P , is the BER of user i at the corresponding GR output; 10 and .
The PCF opt , 1 p can be regarded as the normalized correlation between the true MAI plus the 12 total noise component forming at the GR output and the estimated MAI. Assume that 13 is the data set of all users; 1 given b and  . Following the derivations in 6 [43], the expectation terms with hard decisions in (83) can be evaluated based on Price's 7 theorem [49] as follows 8 is the total background noise variance forming at the GR output taking into account 16 multipath fading channel; 2 w σ is the additive Gaussian noise variance forming at the PF and 17 AF outputs of GR linear tract; the Gaussian Q-function is given by (68). 18 Although numerical integration in [43,56] can be adopted for determining the optimal PCF 19 opt , 1 p for the first stage based on (83)-(90), it requires huge computational complexity. To 20 simplify this problem, we assume that the total background noise forming at the GR output 21 can be considered as a constant factor and may be small enough such that the Q functions 22 in (88) and (90) With this, we can rewrite (88) and (90) as follows: 4 are constants. According to assumptions made above, 7 can be expressed by 8 where 10 With the above results, 6 2 , 1 and 10 3 2 It is interesting to see that the lower and upper boundary values can be explicitly calculated 2 from the processing gain N and the number of users K. 3

Multipath channel 4
Based on representation in (37), we can obtain the received signal vector in the following 5 form: Introduce the following notation for the correlation coefficient 8 and .
In commercial DS-CDMA wireless communication systems, the users' spreading codes are 10 often modulated with another code having a very long period. As far as the received signal 11 is concerned, the spreading code is not periodic. In other words, there will be many possible 12 spreading codes for each user. If we use the result derived above, we then have to calculate 13 the optimum PCFs for each possible code and the computational complexity will become 14 very high. Since the period of the modulating code is usually very long, we can treat the 15 code chips as independent random variables and approximate the correlation coefficient 16 jk  given by (110) as a Gaussian random variable. In this case, the GR output for the first 17 stage can be presented in the following form: 18 where the background noise forming at the GR output is given by (56). 20 Evaluating the GR output process given by (111), based on the well-known results, for 21 example, discussed in [60], we can define the BER performance for the user k in the 22 following form: 23 In (112), we assume that the occurrence of probabilities for are equal, and 1 that the error probabilities for are also equal. As we can see from (111),  2 there are three terms. The first term corresponds to the desired user bit. If we let , it is 3 a deterministic value. The third term in (111) given by (56) where j q is defined in (61), considering jk  as a Gaussian random variable, we obtain and the mathematical expectation of variance as 20 Note that the expectations in (115) and (116) The optimal PCF for the user k can be found as 3 Substituting (115)-(119) into (120) and simplifying the result, we obtain the following 5 equation 6 Unlike that in AWGN channel, the result for the aperiodic code scenario is more difficult to 8 obtain because there are more correlation terms in (114)-(120) to work with. Before 9 evaluation of the expectation terms in (98), we define some function as follows: 10 , , The expectation of jk  over all possible codes can be presented in the following form: In this case, we have 8 At 0   , we have the same result except that the sign of  in (129) is plus. We can conclude 10 that the function ) , , ,   Fig.4. We note 1 the substantial benefits of increasing the number of diversity branches N for fixed L. is presented in Fig. 5. To achieve the same value of average SNR per bit per diversity 12 branch, we should choose 2L quadrature branches for the GR with quadrature subbranch 13 HS/MRC, HS/MRC schemes, and L diversity branches for the GR with traditional HS/MRC 14 scheme. Figure 5 demonstrates that the GR BER performance with quadrature subbranch 15 HS/MRC and HS /MRC schemes is much better than that of the GR with traditional HS/MRC 16 scheme, about 0.5 dB to 1.2 dB, when L is less than one half N. This difference decreases with 17 increasing L. This is expected because when N L  we obtain the same performance. Some 18 discussion of increases in GR complexity and power consumption is in order. We first note 19 that GR with quadrature subbranch HS/MRC and HS/MRC schemes requires the same 20 number of antennas as GR with traditional HS/MRC scheme. On the other hand, the former 21 requires twice as many comparators as the latter, to select the best signals for further 22 processing. However, GR designs that process the quadrature signal components will require 23 2L receiver chains for either the GR with quadrature subbranch HS/MRC and HS/MRC 24 schemes or the GR with traditional HS/MRC scheme. Such receiver designs will use only a 1 little additional power, as GR chains consume much more power than the comparators. 2 3 4 5 On the other hand, GR designs that implement co-phasing of branch signals without 9 splitting the branch signals into the quadrature components will require L receiver chains 10 for GR with traditional HS/MRC scheme and 2L receiver chains for GR with quadrature 11 subbranch HS/MRC and HS/MRC schemes, with corresponding hardware and power 12 consumption increases. 13

Synchronous DS-CDMA wireless communication system 14
To demonstrate a usefulness of the optimal PCF range given by (108)  These results demonstrate us a great superiority of the GR employment over the 7 conventional detector in [43]. 8 Figure 7 shows the BER performance at each stage for the three-stage GR based on the PPIC 9 using different PCFs at the first stage, i.e., the average value and an arbitrary value. PCFs for 10 these two three-stage cases are 11 and multipath channels is also presented in Fig.7. We see that in the case of multipath 19 channel, the BER performance is deteriorated. This fact can be explained by the additional 20 correlation terms in (133) AWGN and for the multipath channels. We carry out simulation for the AWGN channel 2 under the following conditions: the Gold codes, SNR=12 dB, the spreading codes are the 3 periodic and perfect power control. The multipath channel assumed is a two-ray channel 4 with the transfer function 5 for all users. In the case of multipath channel, we employ aperiodic codes, SNR =12 dB, and 7 perfect power control.  values is close to that of the GR of the case using the real optimal PCF, whether the SNR is 22 high or low. It has also been shown that GR employment in DS-CDMA system with 23 multipath fading channel in the case of periodic code scenario allows us to observe a great 24 superiority over the conventional receiver discussed in [43]. The procedure discussed in [43] 1 is also acceptable for GR employment by DS-CDMA systems. It has also been demonstrated 2 that the two-stage GR based on PPIC using the proposed PCF at the first stage achieves such 3 BER performance comparable to that of the three-stage GR based on PPIC using an arbitrary 4 PCF at the first stage. This means that at the same BER performance, the number of stages 5 (or complexity) required for the multistage GR based on PPIC could be reduced when the 6 proposed PCF is used at the first stage. It can be shown that the proposed PCF selection 7 approach is applicable to multipath fading cases at GR employment in DS-CDMA systems 8 even if no perfect power control is assumed but this is a subject of future work. We have 9 also compared the BER performance at the optimal PCF in the case of AWGN and multipath 10 channels and presented a sensitivity of the BER performance to the values of PCF for both 11 cases. 12

Brief review 14
In this section, we consider and study the GR in DS-CDMA wireless communication system 15 with frequency-selective channels. We discuss the linear equalization with the finite impulse 16 response (FIR) beamforming filters and channel estimation and spatially correlation. 17

Linear equalization and FIR beamforming filters 18
The use of multiple antennas in with the eigenvalue μ . Unfortunately, (163) does not seem to have a closed-form solution. 4 Therefore, we use the following gradient algorithm for calculation of the optimum FIR 5 beamforming filters at the GR, which recursively improves an initial beamforming filter 6 vector 0 g . The main statements of the gradient algorithm are: 7 and initialized the beamforming filter vector with a suitable 0 g satisfying 8 2. Update the beamforming filter vector 10 0.5 where i δ is a suitable adaptation step size. 12 3. Normalize the beamforming filter vector 13 Because of non-convexity of the underlying optimization problem, we cannot guarantee that 1 the gradient algorithm converges to the global maximum. However, adopting the 2 initialization procedure explained below, the solution found by this gradient algorithm 3 seems to be close to optimum, i.e., if g L is chosen sufficiently large the FIR beamforming 4 filters obtained with the gradient algorithm approach and the performance of the optimal 5 infinite impulse response beamforming filters at the GR was discussed in [52]. 6 We found empirically that a convergence to the optimum or a close to optimum solution is 7 achieved if the beamforming filter length is gradually increased. If the desired beamforming 8 filter length is g L , the gradient algorithm is executed g L times. The beamforming filter 9 vector is initialized with the normalized all-ones vector of size T N for the first execution 10 of the gradient algorithm. For the υ -th execution, beamforming filters coefficients of each antenna are initialized with the optimum 12 beamforming filter coefficients for that antenna found in the ) 1 (  υ -th execution of the 13 gradient algorithm and the υ -th coefficients are initialized with zero. In each execution step 14 υ , the algorithm requires typically less than 50 iterations to converge, i.e., the overall 15 complexity of the algorithm are on the order of 50 g L . 16

SER definition 11
We continue a discussion of SER formula derivation presented in (61)-(72), subsection 3.3. 1. 12 In the case of M-ary PSK system the exact expression for the SER takes the following form 13 [79] 14 2 0 2 0 Taking into account (61), (67), and (189), we can write the SER of QPSK system employed 16 the GR in the following form: 17 There is a need to note that a direct comparison of QPSK and BPSK systems on the basis of 19 average symbol-energy-to-noise-spectral-density ratio indicates that the QPSK is 20 approximately 3 dB worse than the BPSK. 21 Another signaling scheme that allows multiple signals to be transmitted using quadrature 22 carries is the QAM. In this case, the transmitted signal can be presented in the following 23 form: 24 is the Gaussian Q-function given by 9 For a definition of numerical results using simulation, we consider the typical urban channel 13 [81] of the GSM/EDGE system as a practical example. As is usually done for GSM/EDGE, the 14 transmit pulse shape is modeled as a linearized Gaussian minimum-shift keying pulse [82]. 15 The GR input linear system filter is a square-root raised-cosine filter with roll-off factor 0.3. 16 Furthermore, we assume 3  . The correlation coefficient between all pairs of transmit antennas is 18 Figure 9 shows the average SNR as a function of the SNR noted by for the GR with 20 MMSE linear equalization in the cases of FIR (the curves 2 and 3) and infinite impulse 21 response (the curve 1) beamforming filter, respectively, where b E denotes the average 22 received energy per symbol. The curve 5 corresponds to the case for infinite impulse 23 response beamforming filter for receiver discussed in [67]. The SNR was obtained by 1 averaging the respective SNRs over 1000 independent realizations of the typical urban 2 channel. For this purpose, in the case of FIR beamforming filter at the GR, the SNR given by 3 (159) was used and the corresponding beamforming filters at the GR were calculated using 4 the gradient algorithm discussed in Section 4.3. For infinite impulse response beamforming 5 filter at the GR the result given in [52] was used. 6 7 8 9

12
For comparison, we also show simulation results with FIR linear equalization filters at the 13 GR of length L L F 4  for FIR beamforming filters at the GR with 1  g L (the curve 3) 14 optimized for infinite impulse response linear equalization filters at the GR. These 15 simulation results confirm that sufficiently long FIR linear equalization filters at the GR 16 closely approach performance of infinite impulse response linear equalization filters at the 17 GR, which are necessary for (178) to be valid. As expected, the beamforming with infinite 18 impulse response beamforming filters at the GR constitutes a natural performance upper 19 bound for the beamforming with FIR beamforming filters at the GR. However, interestingly, 20 for the typical urban channel the FIR beamforming filter of length 3  g L (the curve 2) is 21 sufficient to closely approach the performance of the infinite impulse response beamforming 1 at the GR (the curve 1). 2 We note that for high values of even an FIR beamforming filter at the GR of length 3 achieves a performance gain of more than 4.5 dB compared to single antenna 4 (the curve 4). Additional simulations not shown here for 5 other GSM/EDGE channel profiles have shown that, in general, the FIR beamforming filter 6 at the GR of length 6  g L is sufficient to closely approach the performance of the infinite 7 impulse response beamforming at the GR. Thereby, the value of g L required to approach 8 the performance of the infinite impulse response beamforming at the GR seems to be 9 smaller if the channel is less frequency selective. 10

Channel estimation and spatially correlation 11
We choose the 0.5 rate Low Density Parity Check (LDPC) code with a block length of 64800 12 bits, which is also adopted by DVB-S.

Conclusions 4
In the present section, we have considered the FIR beamforming at the GR with perfect 5 channel state information for single carrier transmission over frequency-selective fading 6 channels with zero-forcing linear equalization and GR MMSE linear equalization. We 7 employed a gradient algorithm for efficient recursive calculation of the FIR beamforming 8 filters at the GR. Our results show that for typical GSM/EDGE channel profiles short FIR 9 beamforming filters at the GR suffice to closely approach the performance of optimum 10 infinite impulse response beamforming at the GR discussed in [52]. This is a significant 11 result, since in practice, the quantized beamforming filter coefficients have to be fed back 12 from the receiver to the transmitter, which makes short beamforming filters preferable. 13 The proposed MMSE GR outperforms all the existing schemes with considerable gain 14 especially for receiver correlation MIMO channel scenario. The underlying reason of this 15 improvement is that the MMSE GR, by taking channel estimation error, decision error 16 propagation, and channel correlation into account, can output more reliable LLR to channel 17 decoder. As channel estimation error is the dominant factor influencing the system 18 performance under the lower SNR region, it can observed that the BER of the conventional 19 soft-output MMSE GR [87] is slightly better than that of the modified soft-output MMSE GR 20 in the case of spatially independent MIMO channel. 21 5. Signal processing with fading channels 1

Brief review 2
It is well known that the DS-CDMA transmission technique allows multiple users to share 3 the same spectrum range simultaneously. Using DS-CDMA transmission technique in 4 wireless communication systems, we can reach spectrum efficiency, high system capacity, 5 robustness against interference, high quality of service (QoS) and so on [88,89]. In DS-6 CDMA wireless communication systems, the concatenating orthogonal Walsh-Hadamard 7 (WH) channelization sequences and pseudonoise (PN) random scrambling sequences are 8 used to generate the orthogonal spreading codes employed by multiple users for 9 simultaneous signal transmission. There are multipaths in DS-CDMA wireless 10 communication systems that destroy orthogonality between codes by introducing non-zero 11 time delays and lead to interference among the transmitted signals in the downlink. One 12 way to solve this problem is to use the scrambling sequences with the purpose to randomize 13 signals transmitted by users and inhomogeneous behavior at nonzero time delays.

System model 18
In this section, we think that to assess accurately the effects of multipath fading and 19 interference components from other users on the performance of DS-CDMA downlink 20 wireless communication system it is enough to consider a single-cell environment system 21 model. In particular, we analyze a complex baseband-equivalent model with the binary 22 phase-shift keying (BPSK) data and complex signature sequences over a multipath fading 23 channel for the DS-CDMA downlink wireless communication system. The baseband 24 representation of the total signal transmitted on the downlink can be presented in the 25 following form: 26 where K is the number of users; 28 is the transmitted signal of the k-th user; k a P is the power of the k-th transmitted signal; 30 is the data signal of the k-th user; 2 Throughout this section, we assume that 11 . c T NT  (208) 12 Power control is assumed to be perfect, and we suppose that the transmitted signal power 13 k a P is assumed to be known. We also assume 14 . This is because the periods of these long scrambling 9 codes are much larger than that of the spreading factor N, and have correlation properties 10 similar to those of the random scrambling sequences. 11 For instance, at the base station transmitter in mobile communication system, the signals of 12 all K users are symbol synchronously added before passing through a frequency-selective 13 multipath-fading channel. The complex baseband-equivalent impulse response of the 14 multipath channel can be presented in the following form: Rayleigh-, Rician-, or Nakagami-distributed, depending on a specific channel model. All 19 random variables in (214) are assumed to be independent for l. The channel parameters, 20 such as l α , l θ , and l τ are here assumed to be known in the dispreading and demodulation 21 process, although in practice, the impulse response of the channel is typically estimated 22 using the pilot symbol or pilot channel. 23 Moreover, we assume that the multipaths at the GR input are resolvable and chip-24 synchronized, i.e., they are spaced, at least one chip duration apart in time and the relative 25 delays are multiples of the chip duration. Without loss of generality, the resolved paths are 26 assumed to be numbered such that 27 Hence, the baseband complex representation of the signal at the GR input (the input of GR 29 linear system) of any user is given by 30 is the complex background AWGN with zero mean and one-sided power 2 spectral density 0 N . 3 4 Figure 11. GR structure with 1  L fingers.

5
In order to mitigate the multipath fading effect, the GR with coherent demodulation is 6 implemented. The GR structure is presented in Fig.11, where the number of fingers is equal 7 to the number of resolvable paths. 8 Since the symbols are i.i.d. from one symbol duration to another and from one user to 9 another, without loss of generality, we focus our attention on the GR output of the user 0 for 10 the zero-th transmitted symbol. The complex GR output of the i-th finger of user 0 in 11 accordance with the GASP in noise [1-3,6-9,11] is 12 where k a P is the power of information signal and  k a P is the power of model signal (the 2 reference signal). In practice, we can perform this matching implementing, for example, 3 tracking systems. These statements and possible ways to solve this problem and how we can 4 implement all this in practice are discussed in detail in [1-3,6-9,11]. Thus, the following 5 process, in a general case, is formed at the GR output for k-th user according to 6 implementation of GASP in DS-CDMA wireless communication systems. 7 The case 1: a "yes" signal in the input process -8 where the first term is the signal component; the second term is the multiple-user 1 interference component determined by 2 the third term is the multipath interference component given by 4 (237) 8 is the energy per data symbol of the m-th user and the variance 2 w σ is given by (23). 9 Similarly, when the unified complex Hadamard transform spreading codes are employed by 10 the DS-CDMA downlink wireless communication system using the GR, we can obtain that 11 the multipath interference component Therefore, the SINR at the GR i-th finger output for the half-spectrum property unified 4 complex Hadamard transform spreading sequences employed by the DS-CDMA downlink 5 wireless communication system using the GR is determined by 6 0 2 ( )

GR finger weights 8
There is a need to note that the interference plus the thermal noise power seen by different 9 fingers of the GR employed by DS-CDMA downlink wireless communication systems is 10 different. Assume that the multipath interference signals are uncorrelated from one finger to 11 another. In this case, the optimal weight in terms of the maximizing SINR at the GR output 12 is dependent on the multipath interference. This optimal weight is called the modified 13 maximal ratio combining (MMRC). There is a need to note that for the traditional maximal 14 ratio combining (MRC) the weights are chosen in the following manner 15 spreading sequences and the diversity-combining schemes considered, we adopt a Gaussian 25 approximation approach based on the central limit theorem [99]. The Gaussian 26 approximation is known not only to give accurate estimations of the probability of error in 27 the region of practical interest, but also to offer insights into the effects of various sequence 28 and system parameters and interference sources on the performance of the GR [   where the only finger is selected in the GR. For more fingers and various combining 8 schemes, although simple closed-form bounds for the SEP of the DS-CDMA downlink 9 wireless communication system employing the GR with the WH spreading sequences are 10 difficult to obtain, we believe that the same conclusion can be made under some similar 11 conditions obtained by using Jensen's inequality and some intensive calculations. This 12 conclusion can be verified under discussion of numerical simulations made in the next 13 section. Furthermore, in view of (58) and (64)-(67), we observe that the DS-CDMA downlink 14 wireless communication system employing the GR with the unified complex Hadamard 15 transform spreading sequences can achieve high reliable performance at not only high SNR 16 , as in the case of the Rake receiver, but at low SNR , too. 17

Simulation results 18
In this section, we compare the BER performance of the DS-CDMA downlink wireless  Figure 13 shows the BER performance of the DS-CDMA downlink wireless communication 2 system employing the GR with the unified complex Hadamard transform spreading 3 sequences and WH spreading sequences over different channels, including the AWGN 4 channel, Rayleigh fading channel, and Ricean fading channel with the factor 5  F . It can be 5 seen that in all the channels under consideration, the DS-CDMA downlink wireless 6 communication system employing the GR with the unified complex Hadamard transform 7 spreading sequences has the better BER performance, especially in the high SNR region. 8 Also, we can see a great superiority under employment of the GR in the DS-CDMA 9 downlink wireless communication system with the unified complex Hadamard transform 10 spreading sequences and WH spreading sequences over the implementation of the Rake 11 receiver in the DS-CDMA downlink wireless communication system under the same 12 conditions. 13

Effect of MAI 14
To show the effect of MAI, Fig.14  transform spreading sequences and WH spreading sequences in comparison with 31 employment of the Rake receiver in the same system is evident. These results in simulations 32 have verified our theoretical analysis. 33

Conclusions 34
In this section, we investigate a possibility to employ both the orthogonal unified complex 35 Hadamard transform spreading sequences and the WH real spreading sequences in the DS-36 CDMA downlink wireless communication system using the GR constructed based on the 37 GASP and carry out a comparative analysis. The SINR at the GR output in the DS-CDMA 38 1 Figure 13. BER performance as a function of SNR  spreading sequences is independent of the phase offsets between different paths. The SINR 8 at the GR output in the DS-CDMA downlink wireless communication system with the WH 9 real spreading sequences is a function of the squared cosine of path phase offsets. By this 1 reason, the employment of the orthogonal unified complex Hadamard transform spreading 2 sequences in the DS-CDMA downlink wireless communication system using the GR 3 provides the better BER performance with respect to the DS-CDMA downlink wireless 4 communication system using the GR with the WH real spreading sequences, especially, at 5 high SNRs. Evaluation of the BER performance demonstrating the benefits of employment 6 of the unified complex Hadamard transform spreading sequences and MMRC weights in 7 the GR employed by the DS-CDMA downlink wireless communication system has been 8 carried out over simulation. Comparative analysis of employment of the GR and Rake 9 receiver in the DS-CDMA downlink wireless communication system with the orthogonal 10 unified complex Hadamard transform spreading sequences and the WH real spreading 11 sequences has been performed over simulation. Comparison showed a great superiority in 12 favor of the GR. 13